On a Riemannian manifold with a circulant structure whose third power is the identity

TL;DR

This paper investigates a 3D Riemannian manifold with a circulant tensor structure of order three, exploring its fundamental tensor, conformal invariance, and constructing a Lie group example with specific geometric properties.

Contribution

It introduces a circulant tensor structure of order three on a Riemannian manifold, derives key identities, and constructs a Lie group example with detailed geometric analysis.

Findings

Derived a fundamental tensor identity for the circulant structure.

Showed the tensor identity is preserved under conformal transformations.

Constructed a Lie group example with specific geometric characteristics.

Abstract

It is studied a 3-dimensional Riemannian manifold equipped with a tensor structure of type (1,1), whose third power is the identity. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant. On such a manifold a fundamental tensor by the metric and by the covariant derivative of the circulant structure is defined. An important characteristic identity for this tensor is obtained. It is established that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity. A Lie group as a manifold of the considered type is constructed and some of its geometrical characteristics are found.